The Matrix Cookbook. [ ] Kaare Brandt Petersen Michael Syskind Pedersen. Version: September 5, 2007

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1 The Matrix Cookbook ] Kaare Brandt Petersen Michael Syskind Pedersen Version: September 5, 007 What is this? These pages are a collection of facts (identities, approximations, inequalities, relations,...) about matrices and matters relating to them. It is collected in this form for the convenience of anyone who wants a quick desktop reference. Disclaimer: The identities, approximations and relations presented here were obviously not invented but collected, borrowed and copied from a large amount of sources. These sources include similar but shorter notes found on the internet and appendices in books - see the references for a full list. Errors: Very likely there are errors, typos, and mistakes for which we apologize and would be grateful to receive corrections at cookbook@30.dk. Its ongoing: The project of keeping a large repository of relations involving matrices is naturally ongoing and the version will be apparent from the date in the header. Suggestions: Your suggestion for additional content or elaboration of some topics is most welcome at cookbook@30.dk. Keywords: Matrix algebra, matrix relations, matrix identities, derivative of determinant, derivative of inverse matrix, differentiate a matrix. Acknowledgements: We would like to thank the following for contributions and suggestions: Bill Baxter, Christian Rishøj, Douglas L. Theobald, Esben Hoegh-Rasmussen, Jan Larsen, Korbinian Strimmer, Lars Christiansen, Lars Kai Hansen, Leland Wilkinson, Liguo He, Loic Thibaut, Ole Winther, Stephan Hattinger, and Vasile Sima. We would also like thank The Oticon Foundation for funding our PhD studies. 1

2 CONTENTS CONTENTS Contents 1 Basics Trace and Determinants The Special Case x Derivatives 7.1 Derivatives of a Determinant Derivatives of an Inverse Derivatives of Eigenvalues Derivatives of Matrices, Vectors and Scalar Forms Derivatives of Traces Derivatives of vector norms Derivatives of matrix norms Derivatives of Structured Matrices Inverses Basic Exact Relations Implication on Inverses Approximations Generalized Inverse Pseudo Inverse Complex Matrices 4.1 Complex Derivatives Decompositions Eigenvalues and Eigenvectors Singular Value Decomposition Triangular Decomposition Statistics and Probability Definition of Moments Expectation of Linear Combinations Weighted Scalar Variable Multivariate Distributions Student s t Cauchy Gaussian Multinomial Dirichlet Normal-Inverse Gamma Wishart Inverse Wishart Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page

3 CONTENTS CONTENTS 8 Gaussians Basics Moments Miscellaneous Mixture of Gaussians Special Matrices Orthogonal, Ortho-symmetric, and Ortho-skew Units, Permutation and Shift The Singleentry Matrix Symmetric and Antisymmetric Orthogonal matrices Vandermonde Matrices Toeplitz Matrices The DFT Matrix Positive Definite and Semi-definite Matrices Block matrices Functions and Operators Functions and Series Kronecker and Vec Operator Solutions to Systems of Equations Vector Norms Matrix Norms Rank Integral Involving Dirac Delta Functions Miscellaneous A One-dimensional Results 59 A.1 Gaussian A. One Dimensional Mixture of Gaussians B Proofs and Details 6 B.1 Misc Proofs Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 3

4 CONTENTS CONTENTS Notation and Nomenclature A Matrix A ij Matrix indexed for some purpose A i Matrix indexed for some purpose A ij Matrix indexed for some purpose A n Matrix indexed for some purpose or The n.th power of a square matrix A 1 The inverse matrix of the matrix A A + The pseudo inverse matrix of the matrix A (see Sec. 3.6) A 1/ The square root of a matrix (if unique), not elementwise (A) ij The (i, j).th entry of the matrix A A ij The (i, j).th entry of the matrix A A] ij The ij-submatrix, i.e. A with i.th row and j.th column deleted a Vector a i Vector indexed for some purpose a i The i.th element of the vector a a Scalar Rz Rz RZ Iz Iz IZ Real part of a scalar Real part of a vector Real part of a matrix Imaginary part of a scalar Imaginary part of a vector Imaginary part of a matrix det(a) Determinant of A Tr(A) Trace of the matrix A diag(a) Diagonal matrix of the matrix A, i.e. (diag(a)) ij = δ ij A ij vec(a) The vector-version of the matrix A (see Sec. 10..) sup Supremum of a set A Matrix norm (subscript if any denotes what norm) A T Transposed matrix A Complex conjugated matrix A H Transposed and complex conjugated matrix (Hermitian) A B A B Hadamard (elementwise) product Kronecker product 0 The null matrix. Zero in all entries. I The identity matrix The single-entry matrix, 1 at (i, j) and zero elsewhere Σ A positive definite matrix Λ A diagonal matrix J ij Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 4

5 1 BASICS 1 Basics (AB) 1 = B 1 A 1 (1) (ABC...) 1 =...C 1 B 1 A 1 () (A T ) 1 = (A 1 ) T (3) (A + B) T = A T + B T (4) (AB) T = B T A T (5) (ABC...) T =...C T B T A T (6) (A H ) 1 = (A 1 ) H (7) (A + B) H = A H + B H (8) (AB) H = B H A H (9) (ABC...) H =...C H B H A H (10) 1.1 Trace and Determinants Tr(A) = i A ii (11) Tr(A) = i λ i, λ i = eig(a) (1) Tr(A) = Tr(A T ) (13) Tr(AB) = Tr(BA) (14) Tr(A + B) = Tr(A) + Tr(B) (15) Tr(ABC) = Tr(BCA) = Tr(CAB) (16) det(a) = i λ i λ i = eig(a) (17) det(ca) = c n det(a), if A R n n (18) det(ab) = det(a) det(b) (19) det(a 1 ) = 1/ det(a) (0) det(a n ) = det(a) n (1) det(i + uv T ) = 1 + u T v () det(i + εa) = 1 + εtr(a), ε small (3) 1. The Special Case x Consider the matrix A Determinant and trace A = ] A11 A 1 A 1 A det(a) = A 11 A A 1 A 1 (4) Tr(A) = A 11 + A (5) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 5

6 1. The Special Case x 1 BASICS Eigenvalues λ λ Tr(A) + det(a) = 0 λ 1 = Tr(A) + Tr(A) 4 det(a) λ 1 + λ = Tr(A) λ = Tr(A) Tr(A) 4 det(a) λ 1 λ = det(a) Eigenvectors v 1 ] A 1 λ 1 A 11 v ] A 1 λ A 11 Inverse A 1 = det(a) A 1 A 11 1 A A 1 ] (6) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 6

7 DERIVATIVES Derivatives This section is covering differentiation of a number of expressions with respect to a matrix X. Note that it is always assumed that X has no special structure, i.e. that the elements of X are independent (e.g. not symmetric, Toeplitz, positive definite). See section.8 for differentiation of structured matrices. The basic assumptions can be written in a formula as X kl X ij = δ ik δ lj (7) that is for e.g. vector forms, ] x = x i y y i ] x = x y i y i ] x = x i y ij y j The following rules are general and very useful when deriving the differential of an expression (18]): A = 0 (A is a constant) (8) (αx) = αx (9) (X + Y) = X + Y (30) (Tr(X)) = Tr(X) (31) (XY) = (X)Y + X(Y) (3) (X Y) = (X) Y + X (Y) (33) (X Y) = (X) Y + X (Y) (34) (X 1 ) = X 1 (X)X 1 (35) (det(x)) = det(x)tr(x 1 X) (36) (ln(det(x))) = Tr(X 1 X) (37) X T = (X) T (38) X H = (X) H (39).1 Derivatives of a Determinant.1.1 General form det(y) x = det(y)tr Y 1 Y x ] (40).1. Linear forms det(x) X det(axb) X = det(x)(x 1 ) T (41) = det(axb)(x 1 ) T = det(axb)(x T ) 1 (4) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 7

8 . Derivatives of an Inverse DERIVATIVES.1.3 Square forms If X is square and invertible, then det(x T AX) X If X is not square but A is symmetric, then det(x T AX) X If X is not square and A is not symmetric, then det(x T AX) X.1.4 Other nonlinear forms Some special cases are (See 9, 7]) = det(x T AX)X T (43) = det(x T AX)AX(X T AX) 1 (44) = det(x T AX)(AX(X T AX) 1 + A T X(X T A T X) 1 ) (45) ln det(x T X) X = (X + ) T (46) ln det(x T X) X + = X T (47) ln det(x) X = (X 1 ) T = (X T ) 1 (48) det(x k ) X = k det(x k )X T (49). Derivatives of an Inverse From 6] we have the basic identity from which it follows Y 1 x Y = Y 1 x Y 1 (50) (X 1 ) kl X ij = (X 1 ) ki (X 1 ) jl (51) a T X 1 b X det(x 1 ) X Tr(AX 1 B) X = X T ab T X T (5) = det(x 1 )(X 1 ) T (53) = (X 1 BAX 1 ) T (54) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 8

9 .3 Derivatives of Eigenvalues DERIVATIVES.3 Derivatives of Eigenvalues eig(x) = Tr(X) = I (55) X X eig(x) = X X det(x) = det(x)x T (56).4 Derivatives of Matrices, Vectors and Scalar Forms.4.1 First Order x T a x a T Xb X a T X T b X a T Xa X.4. Second Order X ij = at x x = a (57) = ab T (58) = ba T (59) = at X T a X = aa T (60) X X ij = J ij (61) (XA) ij X mn = δ im (A) nj = (J mn A) ij (6) (X T A) ij X mn = δ in (A) mj = (J nm A) ij (63) X kl (64) X kl X mn klmn = kl b T X T Xc X = X(bc T + cb T ) (65) (Bx + b) T C(Dx + d) x = B T C(Dx + d) + D T C T (Bx + b) (66) (X T BX) kl X ij = δ lj (X T B) ki + δ kj (BX) il (67) (X T BX) X ij = X T BJ ij + J ji BX (J ij ) kl = δ ik δ jl (68) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 9

10 .4 Derivatives of Matrices, Vectors and Scalar Forms DERIVATIVES See Sec 9.3 for useful properties of the Single-entry matrix J ij x T Bx x = (B + B T )x (69) b T X T DXc X = D T Xbc T + DXcb T (70) X (Xb + c)t D(Xb + c) = (D + D T )(Xb + c)b T (71) Assume W is symmetric, then s (x As)T W(x As) = A T W(x As) (7) s (x s)t W(x s) = W(x s) (73) x (x As)T W(x As) = W(x As) (74) A (x As)T W(x As) = W(x As)s T (75).4.3 Higher order and non-linear For proof of the above, see B.1.1. (X n n 1 ) kl = (X r J ij X n 1 r ) kl (76) X ij r=0 n 1 X at X n b = (X r ) T ab T (X n 1 r ) T (77) r=0 X at (X n ) T X n b = n 1 X n 1 r ab T (X n ) T X r r=0 +(X r ) T X n ab T (X n 1 r ) T ] (78) See B.1.1 for a proof. Assume s and r are functions of x, i.e. s = s(x), r = r(x), and that A is a constant, then x st Ar = ] T s Ar + x ] T r A T s (79) x Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 10

11 .5 Derivatives of Traces DERIVATIVES.4.4 Gradient and Hessian Using the above we have for the gradient and the hessian.5 Derivatives of Traces.5.1 First Order f = x T Ax + b T x (80) x f = f x = (A + AT )x + b (81) f xx T = A + A T (8) Tr(X) X = I (83) X Tr(XA) = AT (84) X Tr(AXB) = AT B T (85) X Tr(AXT B) = BA (86) X Tr(XT A) = A (87) X Tr(AXT ) = A (88) Tr(A X) X = Tr(A)I (89) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 11

12 .5 Derivatives of Traces DERIVATIVES.5. Second Order See 7]. X Tr(X ) = X T (90) X Tr(X B) = (XB + BX) T (91) X Tr(XT BX) = BX + B T X (9) X Tr(XBXT ) = XB T + XB (93) X Tr(AXBX) = AT X T B T + B T X T A T (94) X Tr(XT X) = X (95) X Tr(BXXT ) = (B + B T )X (96) X Tr(BT X T CXB) = C T XBB T + CXBB T (97) X Tr X T BXC ] = BXC + B T XC T (98) X Tr(AXBXT C) = A T C T XB T + CAXB (99) X Tr (AXb + c)(axb + c) T = A T (AXb + c)b T (100) Tr(X X) = Tr(X)Tr(X) = Tr(X)I (101) X X.5.3 Higher Order X Tr(Xk ) = k(x k 1 ) T (10) X Tr(AXk ) = k 1 (X r AX k r 1 ) T (103) r=0 X Tr B T X T CXX T CXB ] = CXX T CXBB T +C T XBB T X T C T X +CXBB T X T CX +C T XX T C T XBB T (104).5.4 Other X Tr(AX 1 B) = (X 1 BAX 1 ) T = X T A T B T X T (105) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 1

13 .6 Derivatives of vector norms DERIVATIVES Assume B and C to be symmetric, then ] X Tr (X T CX) 1 A = (CX(X T CX) 1 )(A + A T )(X T CX) 1 (106) ] X Tr (X T CX) 1 (X T BX) = CX(X T CX) 1 X T BX(X T CX) 1 See 7]. norms og matrix norms..6 Derivatives of vector norms.6.1 Two-norm +BX(X T CX) 1 (107) x x a = x a (108) x a x a I (x a)(x a)t = x x a x a x a 3.7 Derivatives of matrix norms For more on matrix norms, see Sec Frobenius norm See (01). (109) x x = xt x = 1 x x (110) X X F =.8 Derivatives of Structured Matrices X Tr(XXH ) = X (111) Assume that the matrix A has some structure, i.e. symmetric, toeplitz, etc. In that case the derivatives of the previous section does not apply in general. Instead, consider the following general rule for differentiating a scalar function f(a) df = ] ] T f A kl f A = Tr (11) da ij A kl A ij A A ij kl The matrix differentiated with respect to itself is in this document referred to as the structure matrix of A and is defined simply by A A ij = S ij (113) If A has no special structure we have simply S ij = J ij, that is, the structure matrix is simply the singleentry matrix. Many structures have a representation in singleentry matrices, see Sec for more examples of structure matrices. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 13

14 .8 Derivatives of Structured Matrices DERIVATIVES.8.1 The Chain Rule Sometimes the objective is to find the derivative of a matrix which is a function of another matrix. Let U = f(x), the goal is to find the derivative of the function g(u) with respect to X: g(u) X = g(f(x)) X Then the Chain Rule can then be written the following way: (114) g(u) X = g(u) x ij = M N k=1 l=1 Using matrix notation, this can be written as:.8. Symmetric g(u) X ij = Tr ( g(u) U g(u) u kl (115) u kl x ij )T U X ij ]. (116) If A is symmetric, then S ij = J ij + J ji J ij J ij and therefore ] ] T ] df f f f da = + diag A A A (117) That is, e.g., (5]): Tr(AX) X det(x) X ln det(x) X = A + A T (A I), see (11) (118) = det(x)(x 1 (X 1 I)) (119) = X 1 (X 1 I) (10).8.3 Diagonal If X is diagonal, then (18]): Tr(AX) X = A I (11).8.4 Toeplitz Like symmetric matrices and diagonal matrices also Toeplitz matrices has a special structure which should be taken into account when the derivative with respect to a matrix with Toeplitz structure. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 14

15 .8 Derivatives of Structured Matrices DERIVATIVES Tr(AT) T = Tr(TA) T = Tr(A) Tr(A T ] n1 ) Tr(A T ] 1n ] n 1, ) A n1 Tr(A T ] 1n )) Tr(A T ] 1n ],n 1 ) α(a) Tr(A) Tr(A T ]1n ] n 1, ) Tr(A T ]n1 ) A 1n Tr(A T ] 1n ],n 1 ) Tr(A T ] 1n )) Tr(A) (1) As it can be seen, the derivative α(a) also has a Toeplitz structure. Each value in the diagonal is the sum of all the diagonal valued in A, the values in the diagonals next to the main diagonal equal the sum of the diagonal next to the main diagonal in A T. This result is only valid for the unconstrained Toeplitz matrix. If the Toeplitz matrix also is symmetric, the same derivative yields Tr(AT) T = Tr(TA) T = α(a) + α(a) T α(a) I (13) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 15

16 3 INVERSES 3 Inverses 3.1 Basic Definition The inverse A 1 of a matrix A C n n is defined such that AA 1 = A 1 A = I, (14) where I is the n n identity matrix. If A 1 exists, A is said to be nonsingular. Otherwise, A is said to be singular (see e.g. 1]) Cofactors and Adjoint The submatrix of a matrix A, denoted by A] ij is a (n 1) (n 1) matrix obtained by deleting the ith row and the jth column of A. The (i, j) cofactor of a matrix is defined as cof(a, i, j) = ( 1) i+j det(a] ij ), (15) The matrix of cofactors can be created from the cofactors cof(a, 1, 1) cof(a, 1, n) cof(a) =. cof(a, i, j). cof(a, n, 1) cof(a, n, n) The adjoint matrix is the transpose of the cofactor matrix (16) adj(a) = (cof(a)) T, (17) Determinant The determinant of a matrix A C n n is defined as (see 1]) Construction det(a) = = n ( 1) j+1 A 1j det (A] 1j ) (18) j=1 n A 1j cof(a, 1, j). (19) j=1 The inverse matrix can be constructed, using the adjoint matrix, by A 1 = For the case of matrices, see section adj(a) (130) det(a) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 16

17 3. Exact Relations 3 INVERSES Condition number The condition number of a matrix c(a) is the ratio between the largest and the smallest singular value of a matrix (see Section 5. on singular values), c(a) = d + d (131) The condition number can be used to measure how singular a matrix is. If the condition number is large, it indicates that the matrix is nearly singular. The condition number can also be estimated from the matrix norms. Here c(a) = A A 1, (13) where is a norm such as e.g the 1-norm, the -norm, the -norm or the Frobenius norm (see Sec 10.5 for more on matrix norms). The -norm of A equals (max(eig(a H A))) 1, p.57]. For a symmetric matrix, this reduces to A = max( eig(a) ) 1, p.394]. If the matrix ia symmetric and positive definite, A = max(eig(a)). The condition number based on the -norm thus reduces to A A 1 = max(eig(a)) max(eig(a 1 )) = max(eig(a)) min(eig(a)). (133) 3. Exact Relations 3..1 Basic 3.. The Woodbury identity (AB) 1 = B 1 A 1 (134) The Woodbury identity comes in many variants. The latter of the two can be found in 1] (A + CBC T ) 1 = A 1 A 1 C(B 1 + C T A 1 C) 1 C T A 1 (135) (A + UBV) 1 = A 1 A 1 U(B 1 + VA 1 U) 1 VA 1 (136) If P, R are positive definite, then (see 9]) (P 1 + B T R 1 B) 1 B T R 1 = PB T (BPB T + R) 1 (137) 3..3 The Kailath Variant See 4, page 153]. (A + BC) 1 = A 1 A 1 B(I + CA 1 B) 1 CA 1 (138) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 17

18 3. Exact Relations 3 INVERSES 3..4 The Searle Set of Identities The following set of identities, can be found in 4, page 151], (I + A 1 ) 1 = A(A + I) 1 (139) (A + BB T ) 1 B = A 1 B(I + B T A 1 B) 1 (140) (A 1 + B 1 ) 1 = A(A + B) 1 B = B(A + B) 1 A (141) A A(A + B) 1 A = B B(A + B) 1 B (14) A 1 + B 1 = A 1 (A + B)B 1 (143) (I + AB) 1 = I A(I + BA) 1 B (144) (I + AB) 1 A = A(I + BA) 1 (145) 3..5 Rank-1 update of Moore-Penrose Inverse The following is a rank-1 update for the Moore-Penrose pseudo-inverse and proof can be found in 17]. The matrix G is defined below: Using the the notation (A + cd T ) + = A + + G (146) β = 1 + d T A + c (147) v = A + c (148) n = (A + ) T d (149) w = (I AA + )c (150) m = (I A + A) T d (151) the solution is given as six different cases, depending on the entities w, m, and β. Please note, that for any (column) vector v it holds that v + = v T (v T v) 1 = vt v. The solution is: Case 1 of 6: If w = 0 and m = 0. Then G = vw + (m + ) T n T + β(m + ) T w + (15) = 1 w vwt 1 β m mnt + m w mwt (153) Case of 6: If w = 0 and m = 0 and β = 0. Then G = vv + A + (m + ) T n T (154) = 1 v vvt A + 1 m mnt (155) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 18

19 3.3 Implication on Inverses 3 INVERSES Case 3 of 6: If w = 0 and β 0. Then G = 1 ( ) ( ) β mvt A + β v m T v m + β β m + v β (A+ ) T v + n (156) Case 4 of 6: If w = 0 and m = 0 and β = 0. Then G = A + nn + vw + (157) = 1 n A+ nn T 1 w vwt (158) Case 5 of 6: If m = 0 and β 0. Then G = 1 β A+ nw T β n w + β ( w Case 6 of 6: If w = 0 and m = 0 and β = 0. Then β ) ( ) n T A+ n + v β w + n (159) G = vv + A + A + nn + + v + A + nvn + (160) = 1 v vvt A + 1 n A+ nn T + vt A + n v n vnt (161) 3.3 Implication on Inverses See 4]. If (A + B) 1 = A 1 + B 1 then AB 1 A = BA 1 B (16) A PosDef identity Assume P, R to be positive definite and invertible, then See 9]. (P 1 + B T R 1 B) 1 B T R 1 = PB T (BPB T + R) 1 (163) 3.4 Approximations The following is a Taylor expansion (I + A) 1 = I A + A A (164) The following approximation is from 1] and holds when A large and symmetric If σ is small compared to Q and M then A A(I + A) 1 A = I A 1 (165) (Q + σ M) 1 = Q 1 σ Q 1 MQ 1 (166) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 19

20 3.5 Generalized Inverse 3 INVERSES 3.5 Generalized Inverse Definition A generalized inverse matrix of the matrix A is any matrix A such that (see 5]) AA A = A (167) The matrix A is not unique. 3.6 Pseudo Inverse Definition The pseudo inverse (or Moore-Penrose inverse) of a matrix A is the matrix A + that fulfils I AA + A = A II A + AA + = A + III IV AA + symmetric A + A symmetric The matrix A + is unique and does always exist. Note that in case of complex matrices, the symmetric condition is substituted by a condition of being Hermitian Properties Assume A + to be the pseudo-inverse of A, then (See 3]) Assume A to have full rank, then Construction (A + ) + = A (168) (A T ) + = (A + ) T (169) (ca) + = (1/c)A + (170) (A T A) + = A + (A T ) + (171) (AA T ) + = (A T ) + A + (17) (AA + )(AA + ) = AA + (173) (A + A)(A + A) = A + A (174) Assume that A has full rank, then Tr(AA + ) = rank(aa + ) (See 5]) (175) Tr(A + A) = rank(a + A) (See 5]) (176) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 0

21 3.6 Pseudo Inverse 3 INVERSES A n n Square rank(a) = n A + = A 1 A n m Broad rank(a) = n A + = A T (AA T ) 1 A n m Tall rank(a) = m A + = (A T A) 1 A T Assume A does not have full rank, i.e. A is n m and rank(a) = r < min(n, m). The pseudo inverse A + can be constructed from the singular value decomposition A = UDV T, by A + = V r D 1 r U T r (177) where U r, D r, and V r are the matrices with the degenerated rows and columns deleted. A different way is this: There do always exist two matrices C n r and D r m of rank r, such that A = CD. Using these matrices it holds that See 3]. A + = D T (DD T ) 1 (C T C) 1 C T (178) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 1

22 4 COMPLEX MATRICES 4 Complex Matrices 4.1 Complex Derivatives In order to differentiate an expression f(z) with respect to a complex z, the Cauchy-Riemann equations have to be satisfied (7]): df(z) dz and df(z) dz or in a more compact form: = R(f(z)) Rz = i R(f(z)) Iz + i I(f(z)) Rz + I(f(z)) Iz (179) (180) f(z) Iz = if(z) Rz. (181) A complex function that satisfies the Cauchy-Riemann equations for points in a region R is said yo be analytic in this region R. In general, expressions involving complex conjugate or conjugate transpose do not satisfy the Cauchy-Riemann equations. In order to avoid this problem, a more generalized definition of complex derivative is used (3], 6]): Generalized Complex Derivative: df(z) dz = 1 Conjugate Complex Derivative ( f(z) ) Rz if(z). (18) Iz df(z) dz = 1 ( f(z) ) Rz + if(z). (183) Iz The Generalized Complex Derivative equals the normal derivative, when f is an analytic function. For a non-analytic function such as f(z) = z, the derivative equals zero. The Conjugate Complex Derivative equals zero, when f is an analytic function. The Conjugate Complex Derivative has e.g been used by 0] when deriving a complex gradient. Notice: df(z) dz f(z) Rz + if(z) Iz. (184) Complex Gradient Vector: If f is a real function of a complex vector z, then the complex gradient vector is given by (14, p. 798]) f(z) = df(z) dz (185) = f(z) Rz + if(z) Iz. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page

23 4.1 Complex Derivatives 4 COMPLEX MATRICES Complex Gradient Matrix: If f is a real function of a complex matrix Z, then the complex gradient matrix is given by (]) f(z) = df(z) dz (186) = f(z) RZ + if(z) IZ. These expressions can be used for gradient descent algorithms The Chain Rule for complex numbers The chain rule is a little more complicated when the function of a complex u = f(x) is non-analytic. For a non-analytic function, the following chain rule can be applied (7]) g(u) x = g u u x + g u u x = g u ( g ) u u x + u x (187) Notice, if the function is analytic, the second term reduces to zero, and the function is reduced to the normal well-known chain rule. For the matrix derivative of a scalar function g(u), the chain rule can be written the following way: g(u) X g(u) Tr(( = U )T U) + X 4.1. Complex Derivatives of Traces Tr(( g(u) U ) T U ) X. (188) If the derivatives involve complex numbers, the conjugate transpose is often involved. The most useful way to show complex derivative is to show the derivative with respect to the real and the imaginary part separately. An easy example is: Tr(X ) RX = Tr(XH ) RX i Tr(X ) IX = ) itr(xh IX = I (189) = I (190) Since the two results have the same sign, the conjugate complex derivative (183) should be used. Tr(X) RX = Tr(XT ) RX i Tr(X) IX = ) itr(xt IX = I (191) = I (19) Here, the two results have different signs, and the generalized complex derivative (18) should be used. Hereby, it can be seen that (84) holds even if X is a Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 3

24 4.1 Complex Derivatives 4 COMPLEX MATRICES complex number. Tr(AX H ) RX i Tr(AXH ) IX Tr(AX ) RX i Tr(AX ) IX = A (193) = A (194) = A T (195) = A T (196) Tr(XX H ) RX i Tr(XXH ) IX = Tr(XH X) RX = i Tr(XH X) IX = RX (197) = iix (198) By inserting (197) and (198) in (18) and (183), it can be seen that Tr(XX H ) = X X (199) Tr(XX H ) X = X (00) Since the function Tr(XX H ) is a real function of the complex matrix X, the complex gradient matrix (186) is given by Tr(XX H ) = Tr(XXH ) X = X (01) Complex Derivative Involving Determinants Here, a calculation example is provided. The objective is to find the derivative of det(x H AX) with respect to X C m n. The derivative is found with respect to the real part and the imaginary part of X, by use of (36) and (3), det(x H AX) can be calculated as (see App. B.1. for details) det(x H AX) X and the complex conjugate derivative yields = 1 ( det(x H AX) i det(xh AX) ) RX IX = det(x H AX) ( (X H AX) 1 X H A ) T (0) det(x H AX) X = 1 ( det(x H AX) + i det(xh AX) ) RX IX = det(x H AX)AX(X H AX) 1 (03) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 4

25 5 DECOMPOSITIONS 5 Decompositions 5.1 Eigenvalues and Eigenvectors Definition The eigenvectors v and eigenvalues λ are the ones satisfying Av i = λ i v i (04) AV = VD, (D) ij = δ ij λ i, (05) where the columns of V are the vectors v i 5.1. General Properties Symmetric eig(ab) = eig(ba) (06) A is n m At most min(n, m) distinct λ i (07) rank(a) = r At most r non-zero λ i (08) Assume A is symmetric, then VV T = I (i.e. V is orthogonal) (09) λ i R (i.e. λ i is real) (10) Tr(A p ) = i λp i (11) eig(i + ca) = 1 + cλ i (1) eig(a ci) = λ i c (13) eig(a 1 ) = λ 1 i (14) For a symmetric, positive matrix A, eig(a T A) = eig(aa T ) = eig(a) eig(a) (15) 5. Singular Value Decomposition Any n m matrix A can be written as A = UDV T, (16) where U = eigenvectors of AA T n n D = diag(eig(aa T )) n m V = eigenvectors of A T A m m (17) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 5

26 Assume A R n n. Then A ] = V ] D ] U T ], (19) 5. Singular Value Decomposition 5 DECOMPOSITIONS 5..1 Symmetric Square decomposed into squares Assume A to be n n and symmetric. Then ] ] ] A = V D V T ], (18) where D is diagonal with the eigenvalues of A, and V is orthogonal and the eigenvectors of A. 5.. Square decomposed into squares where D is diagonal with the square root of the eigenvalues of AA T, V is the eigenvectors of AA T and U T is the eigenvectors of A T A Square decomposed into rectangular Assume V D U T = 0 then we can expand the SVD of A into ] ] ] ] D 0 U T A = V V, (0) 0 D where the SVD of A is A = VDU T Rectangular decomposition I Assume A is n m, V is n n, D is n n, U T is n m A ] = V ] D ] U T ], (1) where D is diagonal with the square root of the eigenvalues of AA T, V is the eigenvectors of AA T and U T is the eigenvectors of A T A Rectangular decomposition II Assume A is n m, V is n m, D is m m, U T is m m A ] = V ] D U T () U T 5..6 Rectangular decomposition III Assume A is n m, V is n n, D is n m, U T is m m A ] = V ] D ] U T, (3) where D is diagonal with the square root of the eigenvalues of AA T, V is the eigenvectors of AA T and U T is the eigenvectors of A T A. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 6

27 5.3 Triangular Decomposition 5 DECOMPOSITIONS 5.3 Triangular Decomposition Cholesky-decomposition Assume A is positive definite, then A = B T B, (4) where B is a unique upper triangular matrix. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 7

28 6 STATISTICS AND PROBABILITY 6 Statistics and Probability 6.1 Definition of Moments Assume x R n 1 is a random variable Mean The vector of means, m, is defined by (m) i = x i (5) 6.1. Covariance The matrix of covariance M is defined by (M) ij = (x i x i )(x j x j ) (6) or alternatively as M = (x m)(x m) T (7) Third moments The matrix of third centralized moments in some contexts referred to as coskewness is defined using the notation as m (3) ijk = (x i x i )(x j x j )(x k x k ) (8) M 3 = ] m (3) ::1 m(3) ::...m(3) ::n (9) where : denotes all elements within the given index. M 3 can alternatively be expressed as M 3 = (x m)(x m) T (x m) T (30) Fourth moments The matrix of fourth centralized moments in some contexts referred to as cokurtosis is defined using the notation m (4) ijkl = (x i x i )(x j x j )(x k x k )(x l x l ) (31) as M 4 = ] m (4) ::11 m(4) ::1...m(4) ::n1 m(4) ::1 m(4) ::...m(4) ::n... m(4) ::1n m(4) ::n...m(4) ::nn (3) or alternatively as M 4 = (x m)(x m) T (x m) T (x m) T (33) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 8

29 6. Expectation of Linear Combinations 6 STATISTICS AND PROBABILITY 6. Expectation of Linear Combinations 6..1 Linear Forms Assume X and x to be a matrix and a vector of random variables. Then (see See 5]) EAXB + C] = AEX]B + C (34) VarAx] = AVarx]A T (35) CovAx, By] = ACovx, y]b T (36) Assume x to be a stochastic vector with mean m, then (see 7]) 6.. Quadratic Forms EAx + b] = Am + b (37) EAx] = Am (38) Ex + b] = m + b (39) Assume A is symmetric, c = Ex] and Σ = Varx]. Assume also that all coordinates x i are independent, have the same central moments µ 1, µ, µ 3, µ 4 and denote a = diag(a). Then (See 5]) Ex T Ax] = Tr(AΣ) + c T Ac (40) Varx T Ax] = µ Tr(A ) + 4µ c T A c + 4µ 3 c T Aa + (µ 4 3µ )a T a (41) Also, assume x to be a stochastic vector with mean m, and covariance M. Then (see 7]) E(Ax + a)(bx + b) T ] = AMB T + (Am + a)(bm + b) T (4) Exx T ] = M + mm T (43) Exa T x] = (M + mm T )a (44) Ex T ax T ] = a T (M + mm T ) (45) E(Ax)(Ax) T ] = A(M + mm T )A T (46) E(x + a)(x + a) T ] = M + (m + a)(m + a) T (47) E(Ax + a) T (Bx + b)] = Tr(AMB T ) + (Am + a) T (Bm + b) (48) Ex T x] = Tr(M) + m T m (49) Ex T Ax] = Tr(AM) + m T Am (50) E(Ax) T (Ax)] = Tr(AMA T ) + (Am) T (Am) (51) E(x + a) T (x + a)] = Tr(M) + (m + a) T (m + a) (5) See 7]. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 9

30 6.3 Weighted Scalar Variable 6 STATISTICS AND PROBABILITY 6..3 Cubic Forms Assume x to be a stochastic vector with independent coordinates, mean m, covariance M and central moments v 3 = E(x m) 3 ]. Then (see 7]) E(Ax + a)(bx + b) T (Cx + c)] = Adiag(B T C)v 3 +Tr(BMC T )(Am + a) +AMC T (Bm + b) +(AMB T + (Am + a)(bm + b) T )(Cm + c) Exx T x] = v 3 + Mm + (Tr(M) + m T m)m E(Ax + a)(ax + a) T (Ax + a)] = Adiag(A T A)v 3 +AMA T + (Ax + a)(ax + a) T ](Am + a) +Tr(AMA T )(Am + a) E(Ax + a)b T (Cx + c)(dx + d) T ] = (Ax + a)b T (CMD T + (Cm + c)(dm + d) T ) +(AMC T + (Am + a)(cm + c) T )b(dm + d) T 6.3 Weighted Scalar Variable +b T (Cm + c)(amd T (Am + a)(dm + d) T ) Assume x R n 1 is a random variable, w R n 1 is a vector of constants and y is the linear combination y = w T x. Assume further that m, M, M 3, M 4 denotes the mean, covariance, and central third and fourth moment matrix of the variable x. Then it holds that y = w T m (53) (y y ) = w T M w (54) (y y ) 3 = w T M 3 w w (55) (y y ) 4 = w T M 4 w w w (56) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 30

31 7 MULTIVARIATE DISTRIBUTIONS 7 Multivariate Distributions 7.1 Student s t The density of a Student-t distributed vector t R P 1, is given by ν+p P/ Γ( p(t µ, Σ, ν) = (πν) ) Γ(ν/) det(σ) 1/ 1 + ν 1 (t µ) T Σ 1 (t µ) ] (ν+p )/ (57) where µ is the location, the scale matrix Σ is symmetric, positive definite, ν is the degrees of freedom, and Γ denotes the gamma function. For ν = 1, the Student-t distribution becomes the Cauchy distribution (see sec 7.) Mean E(t) = µ, ν > 1 (58) 7.1. Variance cov(t) = ν Σ, ν > (59) ν Mode The notion mode meaning the position of the most probable value Full Matrix Version mode(t) = µ (60) If instead of a vector t R P 1 one has a matrix T R P N, then the Student-t distribution for T is p(t M, Ω, Σ, ν) = π NP/ P p=1 Γ (ν + P p + 1)/] Γ (ν p + 1)/] ν det(ω) ν/ det(σ) N/ det Ω 1 + (T M)Σ 1 (T M) T] (ν+p )/ (61) where M is the location, Ω is the rescaling matrix, Σ is positive definite, ν is the degrees of freedom, and Γ denotes the gamma function. 7. Cauchy The density function for a Cauchy distributed vector t R P 1, is given by 1+P P/ Γ( p(t µ, Σ) = π ) Γ(1/) det(σ) 1/ 1 + (t µ)t Σ 1 (t µ) ] (1+P )/ (6) where µ is the location, Σ is positive definite, and Γ denotes the gamma function. The Cauchy distribution is a special case of the Student-t distribution. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 31

32 7.3 Gaussian 7 MULTIVARIATE DISTRIBUTIONS 7.3 Gaussian See sec Multinomial If the vector n contains counts, i.e. (n) i 0, 1,,..., then the discrete multinomial disitrbution for n is given by P (n a, n) = n! n 1!... n d! d i a ni i, d n i = n (63) i where a i are probabilities, i.e. 0 a i 1 and i a i = Dirichlet The Dirichlet distribution is a kind of inverse distribution compared to the multinomial distribution on the bounded continuous variate x = x 1,..., x P ] 16, p. 44] ( P ) Γ p α p P p(x α) = P p Γ(α x αp 1 p p) 7.6 Normal-Inverse Gamma 7.7 Wishart The central Wishart distribution for M R P P, M is positive definite, where m can be regarded as a degree of freedom parameter 16, equation 3.8.1] 8, section.5],11] p p(m Σ, m) = 1 mp/ π P (P 1)/4 P p Γ 1 (m + 1 p)] det(σ) m/ det(m) (m P 1)/ exp 1 ] Tr(Σ 1 M) (64) Mean E(M) = mσ (65) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 3

33 7.8 Inverse Wishart 7 MULTIVARIATE DISTRIBUTIONS 7.8 Inverse Wishart The (normal) Inverse Wishart distribution for M R P P, M is positive definite, where m can be regarded as a degree of freedom parameter 11] p(m Σ, m) = 1 mp/ π P (P 1)/4 P p Γ 1 (m + 1 p)] det(σ) m/ det(m) (m P 1)/ exp 1 ] Tr(ΣM 1 ) (66) Mean 1 E(M) = Σ m P 1 (67) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 33

34 8 GAUSSIANS 8 Gaussians 8.1 Basics Density and normalization The density of x N (m, Σ) is 1 p(x) = exp 1 ] det(πσ) (x m)t Σ 1 (x m) (68) Note that if x is d-dimensional, then det(πσ) = (π) d det(σ). Integration and normalization exp 1 ] (x m)t Σ 1 (x m) dx = det(πσ) exp 1 ] xt Ax + b T x dx = ] 1 det(πa 1 ) exp bt A 1 b exp 1 ] Tr(ST AS) + Tr(B T S) ds = ] 1 det(πa 1 ) exp Tr(BT A 1 B) The derivatives of the density are p(x) = p(x)σ 1 (x m) (69) x p ( xx T = p(x) Σ 1 (x m)(x m) T Σ 1 Σ 1) (70) 8.1. Marginal Distribution Assume x N x (µ, Σ) where ] xa x = x b µ = µa µ b ] Σa Σ Σ = c Σ T c Σ b ] (71) then Conditional Distribution p(x a ) = N xa (µ a, Σ a ) (7) p(x b ) = N xb (µ b, Σ b ) (73) Assume x N x (µ, Σ) where ] xa x = x b µ = µa µ b ] Σa Σ Σ = c Σ T c Σ b ] (74) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 34

35 8.1 Basics 8 GAUSSIANS then p(x a x b ) = N xa (ˆµ a, ˆΣ a ) p(x b x a ) = N xb (ˆµ b, ˆΣ b ) { ˆµa = µ a + Σ c Σ 1 b (x b µ b ) ˆΣ a = Σ a Σ c Σ 1 b Σ T (75) c { ˆµb = µ b + Σ T c Σ 1 a (x a µ a ) ˆΣ b = Σ b Σ T c Σ 1 (76) a Σ c Note, that the covariance matrices are the Schur complement of the block matrix, see for details Linear combination Assume x N (m x, Σ x ) and y N (m y, Σ y ) then Ax + By + c N (Am x + Bm y + c, AΣ x A T + BΣ y B T ) (77) Rearranging Means det(π(at Σ N Ax m, Σ] = 1 A) 1 ) N x A 1 m, (A T Σ 1 A) 1 ] (78) det(πσ) Rearranging into squared form If A is symmetric, then 1 xt Ax + b T x = 1 (x A 1 b) T A(x A 1 b) + 1 bt A 1 b 1 Tr(XT AX) + Tr(B T X) = 1 Tr(X A 1 B) T A(X A 1 B)] + 1 Tr(BT A 1 B) Sum of two squared forms In vector formulation (assuming Σ 1, Σ are symmetric) 1 (x m 1) T Σ 1 1 (x m 1) (79) 1 (x m ) T Σ 1 (x m ) (80) = 1 (x m c) T Σ 1 c (x m c ) + C (81) Σ 1 c = Σ Σ 1 (8) m c = (Σ Σ 1 ) 1 (Σ 1 1 m 1 + Σ 1 m ) (83) C = 1 (mt 1 Σ m T Σ 1 )(Σ Σ 1 1 ( ) m T 1 Σ 1 1 m 1 + m T Σ 1 m ) 1 (Σ 1 1 m 1 + Σ 1 m )(84) (85) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 35

36 8. Moments 8 GAUSSIANS In a trace formulation (assuming Σ 1, Σ are symmetric) 1 Tr((X M 1) T Σ 1 1 (X M 1)) (86) 1 Tr((X M ) T Σ 1 (X M )) (87) = 1 Tr(X M c) T Σ 1 c (X M c )] + C (88) Σ 1 c = Σ Σ 1 (89) M c = (Σ Σ 1 ) 1 (Σ 1 1 M 1 + Σ 1 M ) (90) C = 1 ] Tr (Σ 1 1 M 1 + Σ 1 M ) T (Σ Σ 1 ) 1 (Σ 1 1 M 1 + Σ 1 M ) 1 Tr(MT 1 Σ 1 1 M 1 + M T Σ 1 M ) (91) Product of gaussian densities Let N x (m, Σ) denote a density of x, then N x (m 1, Σ 1 ) N x (m, Σ ) = c c N x (m c, Σ c ) (9) c c = N m1 (m, (Σ 1 + Σ )) 1 = det(π(σ1 + Σ )) exp 1 ] (m 1 m ) T (Σ 1 + Σ ) 1 (m 1 m ) m c = (Σ Σ 1 ) 1 (Σ 1 1 m 1 + Σ 1 m ) Σ c = (Σ Σ 1 ) 1 but note that the product is not normalized as a density of x. 8. Moments 8..1 Mean and covariance of linear forms First and second moments. Assume x N (m, Σ) E(x) = m (93) Cov(x, x) = Var(x) = Σ = E(xx T ) E(x)E(x T ) = E(xx T ) mm T (94) As for any other distribution is holds for gaussians that EAx] = AEx] (95) VarAx] = AVarx]A T (96) CovAx, By] = ACovx, y]b T (97) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 36

37 8. Moments 8 GAUSSIANS 8.. Mean and variance of square forms Mean and variance of square forms: Assume x N (m, Σ) E(xx T ) = Σ + mm T (98) Ex T Ax] = Tr(AΣ) + m T Am (99) Var(x T Ax) = σ 4 Tr(A ) + 4σ m T A m (300) E(x m ) T A(x m )] = (m m ) T A(m m ) + Tr(AΣ) (301) Assume x N (0, σ I) and A and B to be symmetric, then Cov(x T Ax, x T Bx) = σ 4 Tr(AB) (30) 8..3 Cubic forms Exb T xx T ] = mb T (M + mm T ) + (M + mm T )bm T 8..4 Mean of Quartic Forms +b T m(m mm T ) (303) Exx T xx T ] = (Σ + mm T ) + m T m(σ mm T ) +Tr(Σ)(Σ + mm T ) Exx T Axx T ] = (Σ + mm T )(A + A T )(Σ + mm T ) +m T Am(Σ mm T ) + TrAΣ](Σ + mm T ) Ex T xx T x] = Tr(Σ ) + 4m T Σm + (Tr(Σ) + m T m) Ex T Axx T Bx] = TrAΣ(B + B T )Σ] + m T (A + A T )Σ(B + B T )m +(Tr(AΣ) + m T Am)(Tr(BΣ) + m T Bm) Ea T xb T xc T xd T x] = (a T (Σ + mm T )b)(c T (Σ + mm T )d) +(a T (Σ + mm T )c)(b T (Σ + mm T )d) +(a T (Σ + mm T )d)(b T (Σ + mm T )c) a T mb T mc T md T m E(Ax + a)(bx + b) T (Cx + c)(dx + d) T ] = AΣB T + (Am + a)(bm + b) T ]CΣD T + (Cm + c)(dm + d) T ] +AΣC T + (Am + a)(cm + c) T ]BΣD T + (Bm + b)(dm + d) T ] +(Bm + b) T (Cm + c)aσd T (Am + a)(dm + d) T ] +Tr(BΣC T )AΣD T + (Am + a)(dm + d) T ] Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 37

38 8.3 Miscellaneous 8 GAUSSIANS E(Ax + a) T (Bx + b)(cx + c) T (Dx + d)] = TrAΣ(C T D + D T C)ΣB T ] +(Am + a) T B + (Bm + b) T A]ΣC T (Dm + d) + D T (Cm + c)] +Tr(AΣB T ) + (Am + a) T (Bm + b)]tr(cσd T ) + (Cm + c) T (Dm + d)] See 7] Moments Ex] = k Cov(x) = k ρ k m k (304) k ρ k ρ k (Σ k + m k m T k m k m T k ) (305) 8.3 Miscellaneous Whitening Assume x N (m, Σ) then z = Σ 1/ (x m) N (0, I) (306) Conversely having z N (0, I) one can generate data x N (m, Σ) by setting x = Σ 1/ z + m N (m, Σ) (307) Note that Σ 1/ means the matrix which fulfils Σ 1/ Σ 1/ = Σ, and that it exists and is unique since Σ is positive definite The Chi-Square connection Assume x N (m, Σ) and x to be n dimensional, then z = (x m) T Σ 1 (x m) χ n (308) where χ n denotes the Chi square distribution with n degrees of freedom Entropy Entropy of a D-dimensional gaussian H(x) = N (m, Σ) ln N (m, Σ)dx = ln det(πσ) + D (309) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 38

39 8.4 Mixture of Gaussians 8 GAUSSIANS 8.4 Mixture of Gaussians Density The variable x is distributed as a mixture of gaussians if it has the density p(x) = K k=1 1 ρ k det(πσk ) exp 1 ] (x m k) T Σ 1 k (x m k) where ρ k sum to 1 and the Σ k all are positive definite Derivatives Defining p(s) = k ρ kn s (µ k, Σ k ) one get ln p(s) ρ j = = ln p(s) µ j = = ln p(s) Σ j = = ρ j N s (µ j, Σ j ) k ρ kn s (µ k, Σ k ) ρ j N s (µ j, Σ j ) k ρ kn s (µ k, Σ k ) ρ j N s (µ j, Σ j ) k ρ kn s (µ k, Σ k ) ρ j N s (µ j, Σ j ) k ρ kn s (µ k, Σ k ) ρ j N s (µ j, Σ j ) lnρ j N s (µ ρ j, Σ j )] j 1 ρ j lnρ j N s (µ µ j, Σ j )] j Σ 1 k (s µ k) ] k ρ lnρ j N s (µ kn s (µ k, Σ k ) Σ j, Σ j )] j ρ j N s (µ j, Σ j ) 1 Σ 1 k ρ j + Σ 1 j kn s (µ k, Σ k ) But ρ k and Σ k needs to be constrained. (s µ j )(s µ j ) T Σ 1 ] j (310) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 39

40 9 SPECIAL MATRICES 9 Special Matrices 9.1 Orthogonal, Ortho-symmetric, and Ortho-skew Orthogonal By definition, the real matrix A is orthogonal if and only if A 1 = A T Basic properties for the orthogonal matrix A A T = A AA T = I A T A = I det(a) = 1 9. Units, Permutation and Shift 9..1 Unit vector Let e i R n 1 be the ith unit vector, i.e. the vector which is zero in all entries except the ith at which it is Rows and Columns 9..3 Permutations i.th row of A = e T i A (311) j.th column of A = Ae j (31) Let P be some permutation matrix, e.g P = = ] e e 1 e 3 = e T e T 1 e T 3 (313) For permutation matrices it holds that and that AP = Ae Ae 1 Ae 3 ] PP T = I (314) PA = e T A e T 1 A e T 3 A (315) That is, the first is a matrix which has columns of A but in permuted sequence and the second is a matrix which has the rows of A but in the permuted sequence. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 40

41 9.3 The Singleentry Matrix 9 SPECIAL MATRICES 9..4 Translation, Shift or Lag Operators Let L denote the lag (or translation or shift ) operator defined on a 4 4 example by L = (316) i.e. a matrix of zeros with one on the sub-diagonal, (L) ij = δ i,j+1. With some signal x t for t = 1,..., N, the n.th power of the lag operator shifts the indices, i.e. { (L n 0 for t = 1,.., n x) t = (317) x t n for t = n + 1,..., N A related but slightly different matrix is the recurrent shifted operator defined on a 4x4 example by ˆL = (318) i.e. a matrix defined by (ˆL) ij = δ i,j+1 + δ i,1 δ j,dim(l). On a signal x it has the effect (ˆL n x) t = x t, t = (t n) mod N] + 1 (319) That is, ˆL is like the shift operator L except that it wraps the signal as if it was periodic and shifted (substituting the zeros with the rear end of the signal). Note that ˆL is invertible and orthogonal, i.e. ˆL 1 = ˆL T (30) 9.3 The Singleentry Matrix Definition The single-entry matrix J ij R n n is defined as the matrix which is zero everywhere except in the entry (i, j) in which it is 1. In a 4 4 example one might have J 3 = (31) The single-entry matrix is very useful when working with derivatives of expressions involving matrices. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 41

42 9.3 The Singleentry Matrix 9 SPECIAL MATRICES 9.3. Swap and Zeros Assume A to be n m and J ij to be m p AJ ij = A i... 0 ] (3) i.e. an n p matrix of zeros with the i.th column of A in place of the j.th column. Assume A to be n m and J ij to be p n 0. 0 J ij A = A j (33) 0. 0 i.e. an p m matrix of zeros with the j.th row of A in the placed of the i.th row Rewriting product of elements A ki B jl = (Ae i e T j B) kl = (AJ ij B) kl (34) A ik B lj = (A T e i e T j B T ) kl = (A T J ij B T ) kl (35) A ik B jl = (A T e i e T j B) kl = (A T J ij B) kl (36) A ki B lj = (Ae i e T j B T ) kl = (AJ ij B T ) kl (37) Properties of the Singleentry Matrix If i = j If i j J ij J ij = J ij (J ij ) T (J ij ) T = J ij J ij (J ij ) T = J ij (J ij ) T J ij = J ij J ij J ij = 0 (J ij ) T (J ij ) T = 0 J ij (J ij ) T = J ii (J ij ) T J ij = J jj The Singleentry Matrix in Scalar Expressions Assume A is n m and J is m n, then Tr(AJ ij ) = Tr(J ij A) = (A T ) ij (38) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 4

43 9.4 Symmetric and Antisymmetric 9 SPECIAL MATRICES Assume A is n n, J is n m and B is m n, then Tr(AJ ij B) = (A T B T ) ij (39) Tr(AJ ji B) = (BA) ij (330) Tr(AJ ij J ij B) = diag(a T B T ) ij (331) Assume A is n n, J ij is n m B is m n, then Structure Matrices The structure matrix is defined by If A has no special structure then x T AJ ij Bx = (A T xx T B T ) ij (33) x T AJ ij J ij Bx = diag(a T xx T B T ) ij (333) A A ij = S ij (334) S ij = J ij (335) If A is symmetric then S ij = J ij + J ji J ij J ij (336) 9.4 Symmetric and Antisymmetric Symmetric The matrix A is said to be symmetric if A = A T (337) Symmetric matrices have many important properties, e.g. that their eigenvalues are real and eigenvectors orthogonal Antisymmetric The antisymmetric matrix is also known as the skew symmetric matrix. It has the following property from which it is defined A = A T (338) Hereby, it can be seen that the antisymmetric matrices always have a zero diagonal. The n n antisymmetric matrices also have the following properties. det(a T ) = det( A) = ( 1) n det(a) (339) det(a) = det( A) = 0, if n is odd (340) The eigenvalues of an antisymmetric matrix are placed on the imaginary axis and the eigenvectors are unitary. Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 43

44 9.5 Orthogonal matrices 9 SPECIAL MATRICES Decomposition A square matrix A can always be written as a sum of a symmetric A + and an antisymmetric matrix A A = A + + A (341) Such a decomposition could e.g. be A = A + AT + A AT = A + + A (34) 9.5 Orthogonal matrices If a square matrix Q is orthogonal Q T Q = QQ T = I. Furthermore Q had the following properties Its eigenvalues are placed on the unit circle. Its eigenvectors are unitary. det(q) = ±1. The inverse of an orthogonal matrix is orthogonal too Ortho-Sym A matrix Q + which simultaneously is orthogonal and symmetric is called an ortho-sym matrix 19]. Hereby Q T +Q + = I (343) Q + = Q T + (344) The powers of an ortho-sym matrix are given by the following rule 9.5. Ortho-Skew Q k + = 1 + ( 1)k = 1 + cos(kπ) I ( 1)k+1 Q + (345) I + 1 cos(kπ) Q + (346) A matrix which simultaneously is orthogonal and antisymmetric is called an ortho-skew matrix 19]. Hereby Q H Q = I (347) Q = Q H (348) The powers of an ortho-skew matrix are given by the following rule Q k = ik + ( i) k I i ik ( i) k Q (349) = cos(k π )I + sin(k π )Q (350) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 44

45 9.6 Vandermonde Matrices 9 SPECIAL MATRICES Decomposition A square matrix A can always be written as a sum of a symmetric A + and an antisymmetric matrix A A = A + + A (351) 9.6 Vandermonde Matrices A Vandermonde matrix has the form 15] V =. 1 v 1 v 1 v n v v v n v n vn vn n 1. (35) The transpose of V is also said to a Vandermonde matrix. The determinant is given by 8] det V = i>j(v i v j ) (353) 9.7 Toeplitz Matrices A Toeplitz matrix T is a matrix where the elements of each diagonal is the same. In the n n square case, it has the following structure: t 11 t 1 t 1n t 0 t 1 t n 1. T = t t1 =. t (354).. t1 t n1 t 1 t 11 t (n 1) t 1 t 0 A Toeplitz matrix is persymmetric. If a matrix is persymmetric (or orthosymmetric), it means that the matrix is symmetric about its northeast-southwest diagonal (anti-diagonal) 1]. Persymmetric matrices is a larger class of matrices, since a persymmetric matrix not necessarily has a Toeplitz structure. There are some special cases of Toeplitz matrices. The symmetric Toeplitz matrix is given by: t 0 t 1 t n 1. T = t (355).. t1 t (n 1) t 1 t 0 Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 45

46 9.8 The DFT Matrix 9 SPECIAL MATRICES The circular Toeplitz matrix: t 0 t 1 t n 1. T C = t..... n t1 t 1 t n 1 t 0 (356) The upper triangular Toeplitz matrix: t 0 t 1 t n 1. T U = t1, (357) 0 0 t 0 and the lower triangular Toeplitz matrix: t T L = t t (n 1) t 1 t Properties of Toeplitz Matrices (358) The Toeplitz matrix has some computational advantages. The addition of two Toeplitz matrices can be done with O(n) flops, multiplication of two Toeplitz matrices can be done in O(n ln n) flops. Toeplitz equation systems can be solved in O(n ) flops. The inverse of a positive definite Toeplitz matrix can be found in O(n ) flops too. The inverse of a Toeplitz matrix is persymmetric. The product of two lower triangular Toeplitz matrices is a Toeplitz matrix. More information on Toeplitz matrices and circulant matrices can be found in 13, 7]. 9.8 The DFT Matrix The DFT matrix is an N N symmetric matrix W N, where the k, nth element is given by = e jπkn N (359) W kn N Thus the discrete Fourier transform (DFT) can be expressed as X(k) = N 1 n=0 x(n)w kn N. (360) Likewise the inverse discrete Fourier transform (IDFT) can be expressed as x(n) = 1 N N 1 k=0 X(k)W kn N. (361) Petersen & Pedersen, The Matrix Cookbook, Version: September 5, 007, Page 46